Symmetric and asymmetric Ramsey properties in random hypergraphs

A celebrated result of R\"odl and Ruci\'nski states that for every graph $F$, which is not a forest of stars and paths of length $3$, and fixed number of colours $r\ge 2$ there exist positive constants $c, C$ such that for $p \leq cn^{-1/m_2(F)}$ the probability that every colouring of the edges of the random graph $G(n,p)$ contains a monochromatic copy of $F$ is $o(1)$ (the "0-statement"), while for $p \geq Cn^{-1/m_2(F)}$ it is $1-o(1)$ (the "1-statement"). Here $m_2(F)$ denotes the $2$-density of $F$. On the other hand, the case where $F$ is a forest of stars has a coarse threshold which is determined by the appearance of a certain small subgraph in $G(n, p)$. Recently, the natural extension of the 1-statement of this theorem to $k$-uniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, R\"odl and Schacht. In particular, they showed an upper bound of order $n^{-1/m_k(F)}$ for the $1$-statement, where $m_k(F)$ denotes the $k$-density of $F$. Similarly as in the graph case, it is known that the threshold for star-like hypergraphs is given by the appearance of small subgraphs. In this paper we show that another type of thresholds exists if $k \ge 4:$ there are $k$-uniform hypergraphs for which the threshold is determined by the asymmetric Ramsey problem in which a different hypergraph has to be avoided in each colour-class. Along the way we obtain a general bound on the $1$-statement for asymmetric Ramsey properties in random hypergraphs. This extends the work of Kohayakawa and Kreuter, and of Kohayakawa, Schacht and Sp\"ohel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.